The first proof, in which we sketch the main argument, is similar to the proof of Proposition 4.1 in [1]}. For a more detailed exposition and much more, see the [2]} — especially Theorem 2.27. A computation similar to that at the end of Section  shows
\( \begin{split}K_{sp} (a, b) &= \sum _{i \ge 1} J_{a+i} (2 \theta ) J_{b+i} (2 \theta ) + \sum _{i \ge 0} J_{a-i} (2 \theta ) J_{b+i} (2 \theta ),\\K_{o} (a, b) &= \sum _{i \ge 0} J_{a+i} (2 \theta ) J_{b+i} (2 \theta ) - \sum _{i \ge 0} J_{a-i} (2 \theta ) J_{b+i} (2 \theta ).\end{split}\) 
